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Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a measure of the amount of rotation an object has, taking into account its mass, shape and speed. It is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. The angular momentum of a system of particles (e.g. a rigid body) is the sum of angular momenta of the individual particles. For a rigid body rotating around an axis of symmetry (e.g. the blades of a ceiling fan), the angular momentum can be expressed as the product of the body's moment of inertia, I'', (i.e., a measure of an object's resistance to changes in its rotation velocity) and its angular velocity, 'ω'''. Tossup Questions # For an electron, the square of this value is equal to h bar squared times the quantum number associated with this quantity times one plus that quantum number. According to the no-hair theorem, a black hole can be completely characterized by mass, charge, and this quantity. This quantity times angular velocity divided by two will give the (*) rotational kinetic energy. It is equal to angular velocity times moment of inertia. For 10 points, name this rotational counterpart of momentum denoted by a capital L. # When two states of this quantity are coupled, Clebsch-Gordan coefficients are used. In the Bohr model, this quantity is given in multiples of h-bar. Kepler's Second Law is a consequence of it being held constant. Spin is a special type of this quantity whose time-derivative is (*) torque. Equal to the product of the moment of inertia and angular velocity, it is symbolized capital L. For 10 points, name this conserved rotational quantity that is the cross-product of position and its linear analogue. # The gyromagnetic ratio of an electron is equal to the magnetic moment divided by this quantity. Its operator is equal to negative i h bar times r cross the gradient. The quantum number j corresponds to the total value of this. Its conservation is a consequence of the isotropy of space. Its derivative with respect to time is torque and Kepler's second law follows from it being conserved. It is equal to the product of moment of inertia and angular velocity. For 10 points, name this rotational version of momentum. # This quantity's interactions are described by the Russell-Saunders coupling scheme. In quantum mechanics, its expansion in uncoupled bases requires Clebsch-Gordan coefficients. One result about this quantity follows from the directional symmetry of space, according to Noether's theorem. A consequence of that result is the fact that a line drawn from a planet to the sun sweeps out equal areas in equal times. Defined as the cross product of the position vector and its linear analogue, this quantity is conserved in the absence of an external torque. For 10 points, name this rotational quantity, the product of moment of inertia and angular velocity. # The quantum coupling of this quantity is described using Clebsch-Gordan coefficients. If a system's Lagrangian is rotationally invariant, this quantity is conserved due to Noether's Theorem. For atomic orbitals, this quantity, represented as the azimuthal quantum number, must be an integer multiple of h bar, and its intrinsic form was quantized in the Stern-Gerlach experiment and is called * spin. Both Kepler's Second Law and gyroscopes demonstrate the conservation of this quantity in the absence of net external torque. For 10 points, name this cross product of position and linear momentum, symbolized with L.